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Existence and nonuniqueness of solutions for a class of asymptotically linear nonperiodic Schrödinger equations

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Abstract

In this paper, we consider the existence and nonuniqueness of solutions for the following Schrödinger equations with the nonlinear term being asymptotically linear at infinity,

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u=f(x,u)&{}\text{ for } x\in {\mathbb {R}}^{N},\\ u(x)\rightarrow 0&{}\text{ as } |x|\rightarrow \infty . \end{array} \right. \end{aligned}$$

We introduce a new condition on V(x) and obtain a new compact embedding theorem. Some new asymptotically linear conditions on f(x, u) are introduced which are quite different from the previous ones in the references. An existence theorem is obtained using the Generalized Mountain Pass theorem. Furthermore, we obtain the existence of infinitely many solutions for above asymptotically linear Schrödinger equations by the Variant Fountain theorem, which has been considered by only few authors.

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Acknowledgements

The authors are very grateful to the referee and the handling editor for valuable comments, which helped us to improve our manuscript greatly.

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Authors and Affiliations

  1. School of Science, Civil Aviation Flight University of China, Guanghan, 618307, People’s Republic of China

    Dong-Lun Wu

  2. The School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Fengying Li

  3. Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu, 610059, Sichuan, People’s Republic of China

    Hongxia Lin

  4. College of Mathematics and Physics, Chengdu University of Technology, Chengdu, 610059, People’s Republic of China

    Hongxia Lin

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  1. Dong-Lun Wu
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Hongxia Lin is supported by the NSF of China (No. 11701049).

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Wu, DL., Li, F. & Lin, H. Existence and nonuniqueness of solutions for a class of asymptotically linear nonperiodic Schrödinger equations. J. Fixed Point Theory Appl. 24, 72 (2022). https://doi.org/10.1007/s11784-022-00975-4

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